Multivariate Quality Control Using Loss-Scaled Principal Components

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Abstract

We consider a principal components based decomposition of the
expected value of the multivariate quadratic loss function, i.e.,
MQL. The principal components are formed by scaling the original
data by the contents of the loss constant matrix, which defines
the economic penalty associated with specific variables being off
their desired target values. We demonstrate the extent to which a
subset of these ``loss-scaled principal components", i.e., LSPC,
accounts for the two components of expected MQL, namely the
trace-covariance term and the off-target vector product. We employ
the LSPC to solve a robust design problem of full and reduced
dimensionality with deterministic models that approximate the true
solution and demonstrate comparable results in less computational
time. We also employ the LSPC to construct a test statistic called
loss-scaled T^2 for multivariate statistical process control.
We show for one case how the proposed test statistic has faster
detection than Hotelling's T^2 of shifts in location for
variables with high weighting in the MQL. In addition we
introduce a principal component based decomposition of Hotelling's
T^2 to diagnose the variables responsible for driving the
location and/or dispersion of a subgroup of multivariate
observations out of statistical control. We demonstrate the
accuracy of this diagnostic technique on a data set from the
literature and show its potential for diagnosing the loss-scaled
T^2 statistic as well.